Home > Articles > Published articles > On McMullen-like mappings |
Date: | 2015 |
Abstract: | We introduce a generalization of the McMullen family f_(z) = z^n /zd^. In 1988 C. McMullen showed that the Julia set of f_ is a Cantor set of circles if and only if 1/n 1/d < 1 and the simple critical values of f_ belong to the trap door. We generalize this behavior and we define a McMullen-like mapping as a rational map f associated to a hyperbolic postcritically finite polynomial P and a pole data D where we encode, basically, the location of every pole of f and the local degree at each pole. In the McMullen family the polynomial P is z z^n and the pole data D is the pole located at the origin that maps to infinity with local degree d. As in the McMullen family f_ we can characterize a McMullen-like mapping using an arithmetic condition depending only on the polynomial P and the pole data D. We prove that the arithmetic condition is necessary using the theory of Thurston's obstructions, and sufficient by quasiconformal surgery. |
Grants: | Ministerio de Economía y Competitividad MTM2008-01486 Agència de Gestió d'Ajuts Universitaris i de Recerca 2009/SGR-792 |
Rights: | Tots els drets reservats. |
Language: | Anglès |
Document: | Article ; recerca ; Versió acceptada per publicar |
Subject: | Complex dynamics ; Julia sets ; McMullen family ; Rational maps |
Published in: | Journal of Fractal Geometry, Vol. 2 (2015) , p. 249-279, ISSN 2308-1317 |
Postprint 21 p, 1.2 MB |